scholarly journals The use of fictitious time in Lagrangian mechanics

2021 ◽  
Vol 18 (2 Jul-Dec) ◽  
pp. 020201
Author(s):  
G. F. Torres del Castillo
Keyword(s):  
Local Time ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Lagrangian Formulation ◽  
Lagrangian Mechanics

We present some examples in the elementary Lagrangian formulation of classical mechanics where the introduction of a parameter in place of the time (sometimes called fictitious time or local time) decouples the equations of motion.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

ROTATIONS AND VIBRATIONS OF FULLERENES IN THE MOLECULAR COMPLEX C20@C80

2021 ◽  
pp. 35-48
Author(s):  
M.A. Bubenchikov ◽  
◽  
A.M. Bubenchikov ◽  
D.V. Mamontov ◽  
◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Rotation Frequency ◽  
Dynamic State ◽  
Large Molecules ◽  
Rotational Degrees Of Freedom ◽  
Rotational Motions ◽  
Centers Of Mass ◽  
The Moment

The aim of this work is to apply classical mechanics to a description of the dynamic state of C20@C80 diamond complex. Endohedral rotations of fullerenes are of great interest due to the ability of the materials created on the basis of onion complexes to accumulate energy at rotational degrees of freedom. For such systems, a concept of temperature is not specified. In this paper, a closed description of the rotation of large molecules arranged in diamond shells is obtained in the framework of the classical approach. This description is used for C20@C80 diamond complex. Two different problems of molecular dynamics, distinguished by a fixing method for an outer shell of the considered bimolecular complex, are solved. In all the cases, the fullerene rotation frequency is calculated. Since a class of possible motions for a single carbon body (molecule) consists of rotations and translational displacements, the paper presents the equations determining each of these groups of motions. Dynamic equations for rotational motions of molecules are obtained employing the moment of momentum theorem for relative motions of the system near the fullerenes’ centers of mass. These equations specify the operation of the complex as a molecular pendulum. The equations of motion of the fullerenes’ centers of mass determine vibrations in the system, i.e. the operation of the complex as a molecular oscillator.

scholarly journals Lagrangian formulation of Newtonian cosmology

2014 ◽  
Vol 36 (3) ◽  
Author(s):  
H.S. Vieira ◽  
V.B. Bezerra
Keyword(s):  
General Relativity ◽  
Differential Equations ◽  
Classical Mechanics ◽  
Lagrangian Formulation ◽  
Einstein Equations ◽  
Lagrangian Formalism ◽  
Newtonian Cosmology

In this paper, we use the Lagrangian formalism of classical mechanics and some assumptions to obtain cosmological differential equations analogous to Friedmann and Einstein equations, obtained from the theory of general relativity. This method can be used to a universe constituted of incoherent matter, that is, the cosmologic substratum is comprised of dust.

scholarly journals VARIATIONAL PROBLEM FOR THE FRENKEL AND THE BARGMANN–MICHEL–TELEGDI (BMT) EQUATIONS

2013 ◽  
Vol 28 (01) ◽  
pp. 1250234 ◽  
Author(s):  
A. A. DERIGLAZOV
Keyword(s):  
Abelian Group ◽  
Variational Problem ◽  
Classical Theory ◽  
Equations Of Motion ◽  
Lagrangian Formulation ◽  
Gauge Invariant ◽  
Local Symmetries

We propose Lagrangian formulation for the particle with value of spin fixed within the classical theory. The Lagrangian is invariant under non-Abelian group of local symmetries. On this reason, all the initial spin variables turn out to be unobservable quantities. As the gauge-invariant variables for description of spin we can take either the Frenkel tensor or the Bargmann–Michel–Telegdi (BMT) vector. Fixation of spin within the classical theory implies O(ℏ)-corrections to the corresponding equations of motion.

Elements of Classical Field Theory

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
Lagrange Equations ◽  
Lie Group Of Transformations ◽  
Infinitesimal Transformations ◽  
Conserved Currents

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

Weyl–Euler–Lagrange equations on twistor space for tangent structure

2016 ◽  
Vol 13 (07) ◽  
pp. 1650095
Author(s):  
Zeki Kasap
Keyword(s):  
Mathematical Modeling ◽  
Dynamical System ◽  
Classical Mechanic ◽  
Twistor Space ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Equation System ◽  
Lagrange Equations ◽  
Four Manifolds ◽  
Riemannian Geometries

Twistor spaces are certain complex three-manifolds, which are associated with special conformal Riemannian geometries on four-manifolds. Also, classical mechanic is one of the major subfields for mechanics of dynamical system. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space for classical mechanic. Euler–Lagrange equations are an efficient use of classical mechanics to solve problems using mathematical modeling. On the other hand, Weyl submitted a metric with a conformal transformation for unified theory of classical mechanic. This paper aims to introduce Euler–Lagrage partial differential equations (mathematical modeling, the equations of motion according to the time) for the movement of objects on twistor space and also to offer a general solution of differential equation system using the Maple software. Additionally, the implicit solution of the equation will be obtained as a result of a special selection of graphics to be drawn.

scholarly journals Lorentz Violation at the Level of Undergraduate Classical Mechanics

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1734 ◽  
Author(s):  
Madeline Clyburn ◽  
Charles D. Lane
Keyword(s):  
Classical Limit ◽  
Binary Star ◽  
Equations Of Motion ◽  
Computational Simulation ◽  
Classical Mechanics ◽  
Lorentz Violation ◽  
Standard Model Extension ◽  
Model Extension ◽  
Binary Star System ◽  
Interesting Behavior

In this paper, we use the classical limit of the Standard-Model Extension to explore some generic features of Lorentz violation. This classical limit is formulated at the level of undergraduate physics. We first discuss the general equations of motion and then concentrate on three specific systems. First, we consider the theoretical aspects of pendulum motion in the presence of Lorentz violation, followed by some sample experimental results. The experimental bounds we achieve, in the range of 10−3, are not competitive with the current bounds from atomic clocks; rather, our experiment illustrates some common ideas and methods that appear in Lorentz-violation studies. We then discuss how Newton’s 2nd Law must be treated with caution in our model. Finally, we introduce a computational simulation of a binary star system that is perturbed by Lorentz-violating effects. This simulation shows some interesting behavior that could be the subject of future analytical studies.

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